Báo cáo hóa học: " Extension of Hu Ke’s inequality and its applications Jing-Feng Tian" pptx

RESEARCH Open Access Extension of Hu Ke’s inequality and its applications Jing-Feng Tian Correspondence: tianjfhxm_ncepu@yahoo.cn College of Science and Technology, North China Electric Power University, Baoding, Hebei Province, 071051, People’s Republic of China Abstract In this paper, we extend Hu Ke’s inequality, which is a sharpness of Hölder’s inequality. Moreover, the obtained results are used to improve Hao Z-C inequality and Beckenbach-type inequ ality that is due to Wang. Mathematics Subject Classification (2000) Primary 26D15; Secondary 26D10 Keywords: integral inequality, Hölder’s inequality, Hu Ke’s inequality, Hao Z-C inequality, Beckenbach-type inequality, arithmetic-geometric mean inequality 1. Introduction The classical Hölder’s inequality states that if a k ≥ 0, b k ≥ 0(k = 1, 2, , n), p >0,q >0 and 1 p + 1 q = 1 , then n  k =1 a k b k ≤  n  k =1 a p k  1 p  n  k =1 b q k  1 q . (1) The inequality (1) is reversed for p <1(p ≠ 0). (For p < 0, we assume that a k , b k > 0.) The following generalization of (1) is given in [1]: Theorem A. ( Generalized Hölder inequality). Let A nj ≥ 0,  n A λ j n j < ∞ , l j >0(j =1, 2, , k). If  k j=1 1 λ j = 1 , then  n k  j =1 A nj ≤ k  j =1   n A λ j nj  1/λ j . (2) As is well known, Hölder’ s inequality plays a very important role in different branches of modern mathematics such as linear algebra, classical r eal and complex analysis, probability and statistics, qualitative theory of differential equations and their applications. A large number of papers dealing with refinements, generalizations and applications of inequalities (1) and (2) and their series analogues in different ares of mathematics have appeared (see e.g. [2-30] and the references therein). Among various refinements of (1), Hu in [ 13] established the following interesting theorems. Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 © 2011 Ti an; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestri cted use, distribution, and reproduction in any medium, provided the original work is properly cited. Theorem B. Let p ≥ q >0, 1 p + 1 q = 1 , let a n , b n ≥ 0,  n a p n < ∞ ,  n b q n < ∞ , and let 1-e n +e m ≥ 0, ∑ n |e n |<∞. Then  n a n b n ≤   n b q n  1 q − 1 p   n b q n   n a p n  2 −   n b q n e n   n a p n  −   n b q n   n a p n e n  2  1 2p . (3) The integral form is as follows: Theorem C. Let E be a measurable set, let f(x) and g(x) be nonnegative measurable functions with ∫ E f p (x)dx < ∞, ∫ E g q (x)dx < ∞, and let e(x) be a measurable function with 1-e(x)+e(y) ≥ 0. If p ≥ q >0, 1 p + 1 q = 1 , then  E f (x)g(x)dx ≤   E g q (x)dx  1 q − 1 p   E f p (x)dx  E g q (x)dx  2 −   E f p (x)e(x)dx  E g q (x)dx −  E f p (x)dx  E g q (x)e(x)dx  2  1 2p . (4) The purpose of t his work is to give extensions of inequalities (3) and (4) and estab- lish their c orresponding reversed versions. M oreover, the obtained results will be applied to improve Hao Z-C inequality [31] and Beckenbach-type inequality that is due to Wang [32]. The rest of this paper is o rganized as follows. In Section 2, we present extensions of (3) and (4) and establish their corresponding reversed versions. In Sec- tion 3, we apply the obtained results to improve Hao Z-C inequality and Beckenbach- type inequality that is due to Wang. Conse quently, we obtain the refinement of arith- metic-geometric mean inequality. Finally, a brief summary is given in Section 4. 2. Extension of Hu Ke’s Inequality We begin this section with two lemmas, which will be used in the sequel. Lemma 2.1. (e.g. [16], p. 12). Let A kj >0(j = 1, 2, , m, k = 1, 2, , n),  m j=1 1 λ j = 1 . If l 1 >0,l j <0(j = 2, 3, , m), then n  k=1 m  j =1 A kj ≥ m  j =1  n  k=1 A λ j kj  1 λ j . (5) Lemma 2.2.[9]If x > -1, a >1ora < 0, then ( 1+x ) α ≥ 1+αx . (6) The inequality is reversed for 0<a <1. Next, we give an extension of Hu Ke’s inequality, as follows. Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page 2 of 14 Theorem 2.3. Let A nj ≥ 0,  n A λ j nj < ∞ (j =1,2, ,k), l 1 ≥ l 2 ≥ ··· ≥ l k >0,  k j=1 1 λ j = 1 , and let 1-e n + e m ≥ 0, ∑ n |e n |<∞. If k is even, then  n k  j=1 A nj ≤ k 2  j=1    n A λ 2j−1 n(2j−1)  1 λ 2j−1 − 1 λ 2j ×    n A λ 2j−1 n(2j−1)   n A λ 2j n(2j)   2 −    n A λ 2j−1 n(2j−1) e n   n A λ 2j n(2j)  −   n A λ 2j−1 n(2j−1)   n A λ 2j n(2j) e n   2  1 2λ 2j  . (7) If k is odd, then  n k  j=1 A nj ≤   n A λ k nk  1 λ k × k − 1 2  j=1    n A λ 2j−1 n(2j−1)  1 λ 2j−1 − 1 λ 2j ×    n A λ 2j−1 n(2j−1)   n A λ 2j n(2j)   2 −    n A λ 2j−1 n(2j−1) e n   n A λ 2j n(2j)  −   n A λ 2j−1 n(2j−1)   n A λ 2j n(2j) e n   2  1 2λ 2j  . (8) The integral form is as follows: Theorem 2.4. Let l 1 ≥ l 2 ≥ ·· · ≥ l k >0,  k j=1 1 λ j = 1 , letEbeameasurableset,F j (x) be nonnegative measurable functions with  E F λ j j (x)dx < ∞ , and let e(x) be a measur - able function with 1-e(x)+e(y) ≥ 0. If k is even, then  E k  j=1 F j (x)dx ≤ k 2  j=1    E F λ 2j−1 2j−1 (x)dx  1 λ 2j−1 − 1 λ 2j ×    E F λ 2j−1 2j−1 (x)dx  E F λ 2j 2j (x)dx  2 −   E F λ 2j−1 2j−1 (x)e(x)dx  E F λ 2j 2j (x)dx −  E F λ 2j−1 2j−1 (x)dx  E F λ 2j 2j (x)e(x)dx  2  1 2λ 2j  . (9) Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page 3 of 14 If k is odd, then  E k  j=1 F j (x)dx ≤   E F λ k k (x)dx  1 λ k × k − 1 2  j=1    E F λ 2j−1 2j−1 (x)dx  1 λ 2j−1 − 1 λ 2j ×    E F λ 2j−1 2j−1 (x)dx  E F λ 2j 2j (x)dx  2 −   E F λ 2j−1 2j−1 (x)e(x)dx  E F λ 2j 2j (x)dx −  E F λ 2j−1 2j−1 (x)dx  E F λ 2j 2j (x)e(x)dx  2  1 2λ 2j  . (10) Proof. We need to prove only Theorem 2.3. The proof of Theorem 2.4 is similar. A simple calculation gives  n  k  j=1 A nj   m  k  i=1 A mi  (1 − e n + e m ) =  n  m  k  j=1 A nj  k  i=1 A mi  −  n  m  k  j=1 A nj  k  i=1 A mi  e n +  n  m  k  j=1 A nj  k  i=1 A mi  e m =   n k  j =1 A nj  2 . (11) Case (I). When k is even, by the inequality (2), we have  n  k  j=1 A nj   m  k  i=1 A mi  (1 − e n + e m ) =  n  k  j=1 A nj   m k  i=1 A mi (1 − e n + e m ) 1 λ i ≤  n  k  j=1 A nj   k  i=1   m A λ i mi (1 − e n + e m )  1 λ i  =  n  k 2  j=1   A λ 2j−1 n(2j−1)  m A λ 2j−1 m(2j−1) (1 − e n + e m )  1 λ 2j−1 − 1 λ 2j ×  A λ 2j−1 n(2j−1)  m A λ 2j m(2j) (1 − e n + e m )  1 λ 2j ×  A λ 2j n(2j)  m A λ 2j−1 m(2j−1) (1 − e n + e m )  1 λ 2j  . (12) Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page 4 of 14 Consequently, a ccording to  1 λ 1 − 1 λ 2  + 1 λ 2 + 1 λ 2 +  1 λ 3 − 1 λ 4  + 1 λ 4 + 1 λ 4 + ···+  1 λ k −1 − 1 λ k  + 1 λ k + 1 λ k = 1 ,by using the inequality (2) on the right side of (12), we observe that  n  k  j=1 A nj   m  k  i=1 A mi  (1 − e n + e m ) ≤ k 2  j=1    n A λ 2j−1 n(2j−1)  m A λ 2j−1 m(2j−1) (1 − e n + e m )  1 λ 2j−1 − 1 λ 2j ×   n A λ 2j−1 n(2j−1)  m A λ 2j m(2j) (1 − e n + e m )  1 λ 2j ×   n A λ 2j n(2j)  m A λ 2j−1 m(2j−1) (1 − e n + e m )  1 λ 2j  = k 2  j=1    n A λ 2j−1 n(2j−1)  2 λ 2j−1 − 2 λ 2j ×   n  m A λ 2j−1 n(2j−1) A λ 2j m(2j) (1 − e n + e m )  ×   n  m A λ 2j n(2j) A λ 2j−1 m(2j−1) (1 − e n + e m )  1 λ 2j  = k 2  j=1    n A λ 2j−1 n(2j−1)  2 λ 2j−1 − 2 λ 2j ×   n A λ 2j−1 n(2j−1)  m A λ 2j m(2j) −  n A λ 2j−1 n(2j−1) e n  m A λ 2j m(2j) +  n A λ 2j−1 n(2j−1)  m A λ 2j m(2j) e m  ×   n A λ 2j n(2j)  m A λ 2j−1 m(2j−1) −  n A λ 2j n(2j) e n  m A λ 2j−1 m(2j−1) +  n A λ 2j n(2j)  m A λ 2j−1 m(2j−1) e m  1 λ 2j  = k 2  j=1    n A λ 2j−1 n(2j−1)  2 λ 2j−1 − 2 λ 2j ×    n A λ 2j−1 n(2j−1)   n A λ 2j n(2j)   2 −    n A λ 2j−1 n(2j−1) e n   n A λ 2j n(2j)  −   n A λ 2j−1 n(2j−1)   n A λ 2j n(2j) e n   2  1 λ 2j  . (13) Combining inequalities (11) and (13) leads to inequality (7) immediately. Case (II). When k is odd, by the same method as in the above case (I), we have the inequality (8). The proof of Theorem 2.3 is complete. □ To illustrate the significance of the introduction of the sequence (e n ) ∞ n = 1 , let us sketch an example as follows. Exampl e 2.5.Let λ j = 1 2 N , j =1,2, ,2N, n =1,2, ,2N, N ≥ 2, let A nj = ⎧ ⎨ ⎩ 1 if j =1, n =1,2, ,2 N 1 if n = j 0 otherwise ,andlet e n =  0 if n eve n 1 if n odd . Then from the generalized Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page 5 of 14 Hölder inequal ity (2), we obtain 0 ≤ ( 2N ) 1 2N . However, from Theorem 2.3, we obtain 0 ≤ 0. Corollary 2.6. Let A nj , l j , e n be as in Theore m 2.3, and let  n A λ j n j = 0 . Then, the fol- lowing inequality holds:  n k  j=1 A nj ≤  k  j=1   n A λ j nj  1 λ j  ×  ρ(k)  j=1  1 − 1 2λ 2j   n A λ 2j−1 n(2j−1) e n  n A λ 2j−1 n ( 2j−1 ) −  n A λ 2j n(2j) e n  n A λ 2j n ( 2j )  2   , (14) where ρ(k)= ⎧ ⎪ ⎨ ⎪ ⎩ k 2 if k eve n k − 1 2 if k odd . Corollary 2 .7. Let F j (x), l j , e(x) be as in Theorem 2.4, and let  E F λ j j (x)dx = 0 . Then, the following inequality holds:  E k  j=1 F j (x)dx ≤  k  j=1   E F λ j j (x)dx  1 λ j  ×  ρ(k)  j=1  1 − 1 2λ 2j   E F λ 2j−1 2j−1 (x)e(x)dx  E F λ 2j−1 2 j −1 (x)dx −  E F λ 2j 2j (x)e(x)dx  E F λ 2j 2 j (x)dx  2   , (15) where ρ(k)= ⎧ ⎪ ⎨ ⎪ ⎩ k 2 if k eve n k − 1 2 if k odd . Proof. We need to prove only Corollary 2.6. The proof of Corollary 2.7 is similar. From inequalities (7) and (8), we obtain  n k  j=1 A nj ≤  k  j=1   n A λ j nj  1 λ j  ×  ρ(k)  j=1  1 −   n A λ 2j−1 n(2j−1) e n  n A λ 2j−1 n ( 2j−1 ) −  n A λ 2j n(2j) e n  n A λ 2j n ( 2j )  2  1 2λ 2j  . (16) Furthermore, performing some simple computations, we have      n A λ 2j−1 n(2j−1) e n  n A λ 2j−1 n ( 2j−1 ) −  n A λ 2j n(2j) e n  n A λ 2j n ( 2j )     < 1 . (17) Consequently, from Lemma 2.2 and the inequalities (16) and (17), we have the desired inequality (14). The proof of Corollary 2.6 is complete. □ It is clear that inequalities (7), (14) and (16) are sharper than the inequality (2). Now, we present the following reversed versions of inequalities (7), (8), (9) and (10). Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page 6 of 14 Theorem 2.8. Let A rj >0,(r = 1, 2, , n, j = 1, 2, , m),  m j=1 1 λ j = 1 , and let 1-e r + e s ≥ 0(s = 1, 2, , n). If l 1 >0,l j <0(j = 2, 3, , m), then n  r=1 m  j=1 A rj ≥  n  r=1 A λ 1 r1  1 λ 1 −  m j=2 1 λ j m  j=2  n  r=1 A λ 1 r1  n  r=1 A λ j rj  2 −  n  r =1 A λ 1 r1 e r  n  r =1 A λ j rj  −  n  r =1 A λ 1 r1  n  r =1 A λ j rj e r  2  1 2λ j . (18) The integral form is as follows: Theorem 2.9. Let F j (x) be nonnegative integrable functions on [a, b] such that  b a F λ j j (x)d x exist, let 1- e(x)+e(y) ≥ 0 for all x, y Î [a, b], and  b a e(x)dx < ∞ , and let  m j=1 1 λ j = 1 . If l 1 >0,l j <0(j = 2, 3, , m), then b  a m  j=1 F j (x)dx ≥  b  a F λ 1 1 (x)dx  1 λ 1 −  m j=2 1 λ j × m  j=2  b  a F λ 1 1 (x)dx b  a F λ j j (x)dx  2 −  b  a F λ 1 1 (x)e(x)dx b  a F λ j j (x)dx − b  a F λ 1 1 (x)dx b  a F λ j j (x)e(x)dx  2  1 2λ j . (19) Proof. We need to p rove only Theorem 2.8. The proof of Theorem 2.9 is similar. By the inequality (5), we have n  s=1  m  i=1 A si  n  r=1  m  j=1 A rj  (1 − e r + e s ) = n  s=1  m  i=1 A si  n  r=1 m  j=1 A rj (1 − e r + e s ) 1 λ j ≥ n  s=1  m  i=1 A si   m  j=1  n  r=1 A λ j rj (1 − e r + e s )  1 λ j  = n  s=1   A λ 1 s1 n  r=1 A λ 1 r1 (1 − e r + e s )  1 λ 1 −  m j=2 1 λ j ×  m  j=2  A λ 1 s1 n  r=1 A λ j rj (1 − e r + e s )  1 λ j  ×  m  j =2  A λ j sj n  r=1 A λ 1 r1 (1 − e r + e s )  1 λ j   . (20) Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page 7 of 14 Consequently, according to  1 λ 1 −  m j=2 1 λ j  + 1 λ 2 + 1 λ 3 + ···+ 1 λ m + 1 λ 2 + 1 λ 3 + ···+ 1 λ m = 1 ,byusingtheinequality (5) on the right side of (20), we observe that n  s=1  m  i=1 A si  n  r=1  m  j=1 A rj  (1 − e r + e s ) ≥  n  s=1 n  r=1 A λ 1 s1 A λ 1 r1 (1 − e r + e s )  1 λ 1 −  m j=2 1 λ j ×  m  j=2  n  s=1 n  r=1 A λ 1 s1 A λ j rj (1 − e r + e s )  1 λ j  ×  m  j=2  n  s=1 n  r=1 A λ j sj A λ 1 r1 (1 − e r + e s )  1 λ j  =  n  r=1 A λ 1 r1  2 λ 1 −  m j=2 2 λ j ×  m  j=2  n  s=1 n  r=1 A λ 1 s1 A λ j rj (1 − e r + e s )  ×  n  s=1 n  r=1 A λ j sj A λ 1 r1 (1 − e r + e s )  1 λ j  =  n  r=1 A λ 1 r1  2 λ 1 −  m j=2 2 λ j ×  m  j=2  n  s=1 A λ 1 s1 n  r=1 A λ j rj − n  s=1 A λ 1 s1 n  r=1 A λ j rj e r + n  s=1 A λ 1 s1 e s n  r=1 A λ j rj  ×  n  s=1 A λ j sj n  r=1 A λ 1 r1 − n  s=1 A λ j sj n  r=1 A λ 1 r1 e r + n  s=1 A λ j sj e s n  r=1 A λ 1 r1  1 λ j  =  n  r=1 A λ 1 r1  2 λ 1 −  m j=2 2 λ j ×  m  j=2   n  r=1 A λ 1 r1  n  r=1 A λ j rj   2 −   n  r=1 A λ 1 r1  n  r=1 A λ j rj e r  −  n  r=1 A λ 1 r1 e r  n  r=1 A λ j rj   2  1 λ j  . (21) Combining inequalities (11) and (21) leads to inequality (18) immediatel y. The proof of Theorem 2.8 is complete. □ Corollary 2.10. Let A rj , l j , e r be as in Theorem 2.8, and let  n r=1 A λ j r j = 0 . Then n  r=1 m  j=1 A rj ≥  m  j=1  n  r=1 A λ j rj  1 λ j  m  j=2  1 − 1 2λ j   n r=1 A λ 1 r1 e r  n r=1 A λ 1 r1 −  n r=1 A λ j rj e r  n r=1 A λ j r j  2   . (22) Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page 8 of 14 Corollary 2.11. Let F j (x), l j , e(x) be as in Theorem 2.9, and let  b a F λ j j (x)dx = 0 . Then b  a m  j=1 F j (x)dx ≥  m  j=1  b  a F λ j j (x)dx  1 λ j  ×  m  j=2  1 − 1 2λ j   b a F λ 1 1 (x)e(x)dx  b a F λ 1 1 (x)dx −  b a F λ j j (x)e(x)dx  b a F λ j j (x)dx  2   . (23) Proof. Making similar arguments as in the proof of Corollary 2.6, we have the desired inequalities (22) and (23). □ It is clea r that in equalities (18) and (22) are sharper than the generalized Hölder inequality (5). Now, we give here some direct consequences from Theorem 2.8 and Theorem 2.9. Putting m = 2 in (18) and (19), respectively, we obtain the following corollaries. Corollary 2.12. Let A r1 , A r2 , l 1 , l 2 , e r be as in Theorem 2.8. Then, the following reversed version of Hu Ke’s inequality (3) holds: n  r=1 A r1 A r2 ≥  n  r=1 A λ 1 r1  1 λ 1 − 1 λ 2  n  r=1 A λ 1 r1  n  r=1 A λ 2 r2  2 −  n  r =1 A λ 1 r1 e r  n  r =1 A λ 2 r2  −  n  r =1 A λ 1 r1  n  r =1 A λ 2 r2 e r  2  1 2λ 2 . (24) Corollary 2.13. Let F 1 (x), F 2 (x), l 1 , l 2 , e(x) be as in Theorem 2.9. Then, the following reversed version of Hu Ke’s inequality (4) holds: b  a F 1 (x)F 2 (x)dx ≥  b  a F λ 1 1 (x)dx  1 λ 1 − 1 λ 2 ×  b  a F λ 1 1 (x)dx b  a F λ 2 2 (x)dx  2 −  b  a F λ 1 1 (x)e(x)dx b  a F λ 2 2 (x)dx − b  a F λ 1 1 (x)dx b  a F λ 2 2 (x)e(x)dx  2  1 2λ 2 . (25) Example 2.14. Putting e(x)= 1 2 cos π(b − x) b − a in (23), we obtain b  a m  j=1 F j (x)dx ≥  m  j=1  b  a F λ j j (x)dx  1 λ j  ×  m  j=2  1 − 1 8λ j   b a F λ 1 1 (x)cos π(b − x) b − a dx  b a F λ 1 1 (x)dx −  b a F λ j j (x)cos π(b − x) b − a dx  b a F λ j j (x)dx  2   , (26) Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page 9 of 14 where l 1 >0,l j <0(j = 2, 3, , m),  m j=1 1 λ j = 1 . 3. Applications In this section, we show some applications of our new inequaliti es. Firstly, we provide an application of the obtained results to improve Hao Z-C inequality, which is related to the generalized arithmetic-geometric mean inequality with weights. The generalized arithmetic-geometric me an inequality (e.g. [9 ]) states that if a j >0,l j >0(j = 1, 2, , k), p > 0 and  k j=1 1 λ j = 1 , then k  j =1 a 1 λ j j ≤ k  j =1 a j λ j . (27) The classical arithmetic-g eometric mean inequality is one of the most important inequalities in analysis. This classical inequality has been widely studied by many authors, and it has motivated a large number of research papers involving different proofs, various generalizations and improvements (see e.g. [1,9,12,19,33] and references therein). In the year 1990, Hao Z-C in [31] established the following interesting inequality k  j=1 a 1 λ j j ≤  p ∞  0  k  j=1 (x + a j ) 1 λ j  −p−1 dx  − 1 p ≤ k  j=1 a j λ j , (28) where a j >0,l j >0(j =1,2, ,k), p >0and  k j=1 1 λ j = 1 . The above Hao Z-C inequality is refined by using Corollary 2.7 as follows: Theorem 3 .1. Let a j >0(j = 1, 2, , k), p >0,let l 1 ≥ l 2 ≥ ··· ≥ l k >0,  k j=1 1 λ j = 1 , and let 1-e(x)+e(y) ≥ 0,  ∞ 0 e(x)dx < ∞ . Then k  j=1 a 1 λ j j ≤  k  j=1 a 1 λ j j  ×  ρ(k)  j=1  1 − 1 2λ j R 2 (x, e; a j , p)   − 1 p ≤  p ∞  0  k  j=1 (x + a j ) 1 λ j  −p−1 dx  − 1 p ≤ k  j=1 a j λ j , (29) where ρ(k)= ⎧ ⎪ ⎨ ⎪ ⎩ k 2 if k eve n k − 1 2 if k odd , R(x, e; a j , p)=  ∞ 0 (x + a 2j−1 ) −p−1 e(x)dx  ∞ 0 (x + a 2j−1 ) −p−1 dx −  ∞ 0 (x + a 2j ) −p−1 e(x)dx  ∞ 0 (x + a 2j ) −p−1 dx . Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page 10 of 14 [...]... Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Combining inequalities (36) and (38) yields inequality (35) The proof of Theorem 3.3 is complete □ 4 Conclusions The classical Hölder’s inequality plays a very important role in both theory and applications In this paper, we have presented an extension of Hu Ke’s inequality, ... (1983) doi:10.1016/0022-247X(83)90126-9 33 Beckenbach, EF: A class of mean-value functions Am Math Monthly 57, 1–6 (1950) doi:10.2307/2305163 doi:10.1186/1029-242X-2011-77 Cite this article as: Tian: Extension of Hu Ke’s inequality and its applications Journal of Inequalities and Applications 2011 2011:77 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer... 2 (1952) 13 Hu, K: On an inequality and its applications Sci Sinica 24, 1047–1055 (1981) 14 Jang, L-C: A note on Hölder type inequality for the fermionic p-adic invariant q-integral J Inequal Appl (2009) 2009, 5 (2009) Article ID 357349 15 Kwon, EG, Bae, EK: On a continuous form of Hölder inequality J Math Anal Appl 343, 585–592 (2008) 16 Kuang, J: Applied Inequalities Shandong Science and Technology... 25 Tian, J: Inequalities and mathematical properties of uncertain variables Fuzzy Optim Decis Ma (2011) 26 Tian, J, Zhang, Z, Tian, D: Moment estimation inequalities based on gλ random variable on Sugeno measure space J Inequal Appl 2010, 10 (2010) Article ID 290124 Page 13 of 14 Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77... any positive numbers a, b, c, the inequality p q a+c b+c T 0 T 0 holds, where 1 hp (x)dx p h(x)g(x)dx ≥ a+c b+c T p 0 f (x)dx T 0 1 p (34) f (x)g(x)dx q ag(x) p The sign of the inequality in (34) is reversed if p > 1 h(x) = ( ) b Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page 12 of 14 Theorem 3.3 Let f(x), g(x),... doi:10.1016/j.aml.2010.05.013 30 Zhu, Y, Liu, B: Some inequalities of random fuzzy variables with application to moment convergence Comput Math Appl 50(5-6), 719–727 (2005) doi:10.1016/j.camwa.2005.04.015 31 Hao, Z-C: Note on the inequality of the arithmetic and geometric means Pacific J Math 143, 43–46 (1990) 32 Wang, C-L: Characteristics of nonlinear positive functionals and their applications J Math Anal...Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page 11 of 14 Proof For x ≥ 0, with a substitution aj ® x + aj in (27), we have 1 (x + aj ) λj ≤ k 0< j=1 k j=1 x + aj =x+ λj k j=1 aj λj (30) Now, integrating both sides of (30) from 0 to ∞, we observe that ∞ 1 (x + a ) λj k −p−1... clear that inequality (33) is sharper than the inequality (27) Now, we give a sharpness of Beckenbach-type inequality from Corollary 2.10 The famous Beckenbach inequality [8] has been generalized and extended in several directions; see, e.g., [16] In 1983, Wang [32] established the following Beckenbach-type inequality Theorem D Let f(x), g(x) be positive integrable functions defined on [0, T], and let... inequality, which is a sharp Hölder’s inequality, and established their corresponding reversed versions Moreover, we have improved Hao Z-C inequality and Beckenbach-type inequality by using the obtained results Finally, we have obtained the refinement of arithmetic-geometric mean inequality We think that our results will be useful for those areas in which inequalities (2) and (5) play a role In the future... Hölder’s inequality Proc Am Math Soc 82, 560–564 (1981) 22 Pang, PYH, Agarwal, RP: On an Opial type inequality due to Fink J Math Anal Appl 196(2), 748–753 (1995) doi:10.1006/ jmaa.1995.1438 23 Pang, PYH, Agarwal, RP: On an integral inequality and its discrete analogue J Math Anal Appl 194(2), 569–577 (1995) doi:10.1006/jmaa.1995.1318 24 Singh, RP, Rajeev, Kumar, Tuteja, RK: Application of Hölder’s inequality . RESEARCH Open Access Extension of Hu Ke’s inequality and its applications Jing-Feng Tian Correspondence: tianjfhxm_ncepu@yahoo.cn College of Science and Technology, North China Electric Power. 1+αx . (6) The inequality is reversed for 0<a <1. Next, we give an extension of Hu Ke’s inequality, as follows. Tian Journal of Inequalities and Applications 2011, 2011:77 http://www.journalofinequalitiesandapplications.com/content/2011/1/77 Page. qualitative theory of differential equations and their applications. A large number of papers dealing with refinements, generalizations and applications of inequalities (1) and (2) and their series